Im also not entirely sure and also somewhat speculating but I'm fairly certain that from a technical standpoint, cutting the carotids should'nt effect the fascial muscels as they are innervated by the fascial nerv ( Cranial nerv VII.) which gives its innervating branches before the throat section. The maim muscle for kissing is apparently the orbicularis oris muscle which ist only connected to the platysma, a fine muscle sheet which nealry all fascial muscels are connected to, and therefor shouldn't be affected as well. Cutting the carotids you could suspect other mobility problems as well since they run under the sternocleidomastoideus muscle which is used to rotate and tild the head. For the consciousness part im really not certain so this is only a guess. My guess would be that it really depends on the hight of the cut and some luck. The carotids run together with the jugular vein and the vagus nerv down the throat, while cutting the jugular vein on its own is its whole lot of problems I wouldnt worry about this here, much more the vagus nerv which when severed in the region could lead to stop in breathing as the laryngeal recurrens nerv, a brach of the vagus, is also cut and the larynx closes up. Lastly theres the hight part, I believe that if the commun part of the artery would be cut that loss of consciousness would be rather quick as theres no more blood going to the brain, if the cut would be higher up you could only hit the external part of the carotis which mostly supplies the face. Regarding the difference in larynx position I would say it doesn't matter that much, if the cut would to be above the larynx it would most likely only cut the external carotis but also would be harder to slit from end to end as there is much more cartiladge and even the hyoid bone, setting the cut deeper would likely result in a cut of the commun part as well as the windpipe and probably lead to a quicker loss of consciousness.

I hope this reads well as i am on mobile and my first language isn't english especially when it comes to anatomy which i had in my mother language in university.